ONT Re: Model Theory

[Jon Awbrey <jawbrey@oakland.edu>]

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Note 23

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| 1.  Introduction
|
| 1.3.  Languages, Models, and Satisfaction (cont.)
|
| A model $A$’ is called a 'submodel' of $A$
| if  $A$’ c $A$ and:
|
|    1.  Each n-placed relation R’ of $A$’
|        is the restriction to  A’ of the
|        corresponding relation R  of $A$,
|        that is:   R’ = R |^| (A’)^n.
|
|    2.  Each m-placed function G’ of $A$’
|        is the restriction to  A’ of the
|        corresponding function G  of $A$,
|        that is:       G’ = G|(A’)^m.
|
|    3.  Each constant of $A$’ is the
|        corresponding constant of $A$.
|
| We use $A$’ ç $A$ to denote that $A$’ is a submodel of $A$, and
| the symbol 'ç' for the submodel relation between models for $L$.
| The reader should show that ç is a partial-order relation and
| that, if $A$ ç $B$, then |A| =< |B|.  We say that $B$ is
| an 'extension' of $A$ if $A$ is a submodel of $B$.
|
| Combining the above two notions, we say that
| $A$ is 'isomorphically embedded' in $B$ if
| there is a model $C$ and an isomorphism f
| such that f : $A$ ~=~ $C$ and $C$ ç $B$.
| In this case we call the function f an
| 'isomorphic embedding' of $A$ in $B$.
| If $A$ is isomorphically embedded
| in $B$, then $B$ is isomorphic
| to an extension of $A$.
|
| Chang & Keisler, 'Model Theory', pages 21-22.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.

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